Theorem intidOLD 5421

Description: Obsolete version of intidg 5420 as of 18-Jan-2025. (Contributed by NM, 5-Jun-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
intid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
intidOLD	⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem intidOLD

Step	Hyp	Ref	Expression
1		snex 5394	. . 3 ⊢ {𝐴} ∈ V
2		eleq2 2818	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
3		intid.1	. . . . 5 ⊢ 𝐴 ∈ V
4	3	snid 4629	. . . 4 ⊢ 𝐴 ∈ {𝐴}
5	2, 4	intmin3 4943	. . 3 ⊢ ({𝐴} ∈ V → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴})
6	1, 5	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴}
7	3	elintab 4925	. . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥))
8		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9	7, 8	mpgbir 1799	. . 3 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
10		snssi 4775	. . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
11	9, 10	ax-mp 5	. 2 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
12	6, 11	eqssi 3966	1 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {cab 2708  Vcvv 3450   ⊆ wss 3917  {csn 4592  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-sn 4593  df-pr 4595  df-int 4914

This theorem is referenced by: (None)